Question 1171196: The human resource manager for a city is evaluating a program designed to improve the health of city office workers by providing an exercise room in all city buildings. The goal is to help city workers lose weight. Before installing the exercise rooms, records indicated that the mean weight of all city office workers was 198 lb. with a standard deviation of 13 lb. A random sample of 40 of these workers taken six months after the exercise rooms were installed showed that the mean of the sample was 192 lb. Construct a 96% confidence interval for the mean weight of all city office employees six months after the program was implemented, assuming the population standard deviation has not changed. Does it appear that the program is working? Explain why or why not.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's construct the confidence interval and analyze the program's effectiveness.
**Given Information:**
* Population mean (μ) before program = 198 lb
* Population standard deviation (σ) = 13 lb
* Sample size (n) = 40
* Sample mean (x̄) = 192 lb
* Confidence level = 96% = 0.96
**1. Find the Critical Value (z-score)**
* Confidence level = 96%
* α = 1 - 0.96 = 0.04
* α/2 = 0.02
* We need to find the z-score that corresponds to a cumulative probability of 1 - 0.02 = 0.98.
* Using a standard normal distribution table or a calculator, we find the z-score (zα/2) ≈ 2.05.
**2. Calculate the Standard Error**
* Standard error (SE) = σ / √n
* SE = 13 / √40
* SE ≈ 13 / 6.3245
* SE ≈ 2.056
**3. Calculate the Margin of Error (ME)**
* Margin of error (ME) = zα/2 * SE
* ME = 2.05 * 2.056
* ME ≈ 4.215
**4. Construct the Confidence Interval**
* Confidence interval = x̄ ± ME
* Confidence interval = 192 ± 4.215
* Confidence interval = (192 - 4.215, 192 + 4.215)
* Confidence interval ≈ (187.785, 196.215)
**5. Interpret the Confidence Interval**
We are 96% confident that the true mean weight of all city office employees six months after the program was implemented is between 187.785 lb and 196.215 lb.
**6. Determine if the Program is Working**
* The confidence interval (187.785, 196.215) is entirely below the original mean weight of 198 lb.
* This suggests that the program may be effective in helping city workers lose weight.
* However, we need to consider the following:
* **Statistical Significance:** We have constructed a confidence interval, but we haven't performed a formal hypothesis test. A hypothesis test would provide a p-value to determine if the weight loss is statistically significant.
* **Practical Significance:** While the confidence interval suggests a decrease in mean weight, we need to consider if this decrease is practically significant. Is a few pounds of weight loss meaningful in terms of overall health?
* **Other Factors:** Weight loss can be influenced by many factors other than the exercise room. It's important to consider if there were other changes during the six months that could have affected weight (e.g., changes in diet, stress levels).
**Conclusion**
Based on the 96% confidence interval, it appears that the program may be working as the interval is below the original mean weight. Further statistical analysis, such as a hypothesis test, and consideration of practical significance and other factors are needed for a more definitive conclusion.
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